In the paper, we introduce the notion of annihilators in BL-algebras and investigate some related properties of them. We get that the ideal lattice (I(L), ⊆) is pseudo-complemented, and for any ideal I, its pseudo-complement is the annihilator I⊥ of I. Also, we define the An (L) to be the set of all annihilators of L, then we have that (An(L); ⋂,∧An(L),⊥,0, L) is a Boolean algebra. In addition, we introduce the annihilators of a nonempty subset X of L with respect to an ideal I and study some properties of them. As an application, we show that if I and J are ideals in a BL-algebra L, then [...] JI⊥ is the relative pseudo-complement of J with respect to I in the ideal lattice (I(L), ⊆). Moreover, we get some properties of the homomorphism image of annihilators, and also give the necessary and sufficient condition of the homomorphism image and the homomorphism pre-image of an annihilator to be an annihilator. Finally, we introduce the notion of α-ideal and give a notation E(I ). We show that (E(I(L)), ∧E, ∨E, E(0), E(L) is a pseudo-complemented lattice, a complete Brouwerian lattice and an algebraic lattice, when L is a BL-chain or a finite product of BL-chains.
@article{bwmeta1.element.doi-10_1515_math-2016-0029,
author = {Yu Xi Zou and Xiao Long Xin and Peng Fei He},
title = {On annihilators in BL-algebras},
journal = {Open Mathematics},
volume = {14},
year = {2016},
pages = {324-337},
zbl = {1349.06034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2016-0029}
}
Yu Xi Zou; Xiao Long Xin; Peng Fei He. On annihilators in BL-algebras. Open Mathematics, Tome 14 (2016) pp. 324-337. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2016-0029/